Re: Is Matrix Subtraction an operation?

Thanks!

I do seem to recall that subtraction is not defined as such for matrices, it is in fact addition of a negative. And that website agrees, but then goes on to do subtraction as an operation ... so that leaves it nicely unresolved.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

Re: Is Matrix Subtraction an operation?

MathsIsFun wrote:

Thanks!

I do seem to recall that subtraction is not defined as such for matrices, it is in fact addition of a negative. And that website agrees, but then goes on to do subtraction as an operation ... so that leaves it nicely unresolved.

Adding a negative and subtraction are different operation that achieve the same thing, so I guess either is okay.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Is Matrix Subtraction an operation?

Hi MathIsFun!

In the field axioms for the real numbers subtraction and division are not mentioned, only addition and multiplication.

Subtraction and Division are INTRODUCED via definitions: x-y is x plus the opposite of y and x/y is x times the multiplicative inverse of y. x-y = x+(-y) where we use "-y" for the opposite of y. So subtraction is a "secondary" operations, not really necessary, but quite handy at times.

But for reciprocals we have no such concise notation. 1/y suggests fractions and uses the symbol"/" that is often interpreted also as division. And x^(-1) is defined typically as 1/y. If we had aSIMPLE notation for RECIPROCALS (such as /x ) then we could define division in a way that would obviously be analogous to the definition of subtraction. x - y = x + (-y) vs x/y = x*(/y)We perhaps get the "-x" from "shortening down 0-x" so that by analogy we could get "/x" from "shortening down 1/x".

But back to linear algebra:Most linear algebra books follow the same pattern for two matrices of the SAME dimensions: A-B is defined as A plus the opposite of B where the opposite of B is the same as B except every entry in -B is the opposite of the corresponding entry in B. So by definition A-B=A+(-B). Some of the books probably just assume the reader understands the "stepping up" of the definition via the field axioms to the situation with matrices.

A similar situation holds for "division" of matrices. If we have the inverse of a matrix A (written A^(-1) ) then we can multiply A*A^(-1) to get an IDENTITY matrix, where the identity matrixfunctions like the multiplicative identity 1 in the reals. But in the reals only zero has no reciprocal.There are many matrices that have no multiplicative inverse, for example, those for which theirdeterminant is zero.

So the choice is up to you whether you want to write A-B vs A+(-B). If one does not allow subtraction then they are stuck with A+(-B).

Have a stupendous day!

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).LaTex is like painting on many strips of paper and then stacking them to see what picture they make.